Nonlocal minimal clusters in the plane

TL;DR

This paper establishes the existence of minimal partitions with fractional perimeter constraints and characterizes their singular cones in the plane, especially near the classical perimeter case.

Contribution

It proves the existence of fractional minimal partitions with Dirichlet conditions and classifies their singular cones in two dimensions for parameters close to one.

Findings

Existence of partitions minimizing fractional perimeter sums.

Unique minimal cone with three phases in weighted cases.

Classification of singular minimal cones near classical perimeter.

Abstract

We prove existence of partitions of an open set $\Omega$ with a given number of phases, which minimize the sum of the fractional perimeters of all the phases, with Dirichlet boundary conditions. In two dimensions we show that, if the fractional parameter $s$ is sufficiently close to $1$, the only singular minimal cone, that is, the only minimal partition invariant by dilations and with a singular point, is given by three half-lines meeting at $120$ degrees. In the case of a weighted sum of fractional perimeters, we show that there exists a unique minimal cone with three phases.